Interacting Many-Body Systems Overview

An interacting many-body system is a quantum system whose Hamiltonian contains genuine interparticle terms or constraints beyond an independent-particle description. A useful schematic form is

$$H = H_0+V,$$

where $H_0$ is a chosen solvable or structurally simple part and $V$ contains interactions. This split is not unique. Calling $V$ “small” is meaningful only after specifying a state, an observable, and an energy or length scale against which it is compared.

Interactions are responsible for much of the distinctive structure of quantum matter: screening, bound states, magnetism, superfluidity, superconductivity, collective modes, Mott physics, and quantum criticality. Yet the mere presence of an interaction does not make a problem uniformly hard. Some weak-coupling problems are destabilized by infrared singularities, while some strong-coupling problems possess a controlled expansion in an inverse coupling or a sharply separated low-energy subspace.

The central task is therefore not to attach the label weak or strong to a dimensional constant. It is to identify the relevant dimensionless controls, compare the competing scales, and choose a representation and method adapted to the observable of interest.

Helpful background. Many-particle Hamiltonians introduces the one- and two-body terms; the thermodynamic limit explains the bulk scaling used below.

Interaction terms and their physical scales

A coupling constant does not specify a regime by itself. The same repulsion $U$ can be small compared with a wide electronic band and large compared with a narrow one. Density, dimension, temperature, particle statistics and the observable being measured also matter.

First identify the degrees of freedom and any constraints, then the spatial and internal structure of their interaction. A contact coupling needs its regulator or low-energy scattering definition; a screened interaction already incorporates part of the medium’s response. An interaction obtained by eliminating a dynamical field can depend on frequency even when the original Hamiltonian is instantaneous.

Pairwise Continuum Interactions

For $N$ identical particles in a continuum, a common first-quantized Hamiltonian is

$$H_N = \sum_{i=1}^{N} \left[ \frac{\mathbf p_i^2}{2m} + U_{\mathrm{ext}}(\mathbf r_i) \right] + \sum_{i<j} v(\mathbf r_i-\mathbf r_j).$$

The restriction $i<j$ counts each unordered pair once. For a translation-invariant, spin-independent interaction, the potential depends only on the relative coordinate. More general interactions can depend on spin, species, orientation, momentum, time, or the center-of-mass position.

For a two-body kernel diagonal in the incoming and outgoing coordinates, the field-operator form is

$$V = \frac12 \int d^d x\,d^d y\, \psi_\alpha^\dagger(\mathbf x) \psi_\beta^\dagger(\mathbf y) v_{\alpha\beta;\gamma\delta} (\mathbf x,\mathbf y) \times \psi_\delta(\mathbf y) \psi_\gamma(\mathbf x),$$

with repeated internal labels summed. The factor $1/2$ again removes pair double counting when the kernel and summation convention are chosen accordingly.

Momentum-dependent or spatially nonlocal two-body kernels require derivatives or four coordinate arguments. This displayed form does not assume weak coupling. It encodes kinematics and pair structure. Whether it is tractable depends on the potential, statistics, density, dimension, state, and observable.

Pairwise is not the same as independent

A pair interaction is a sum of two-particle operators, but the resulting state need not factor into pairs. One particle can correlate with many others through repeated scattering, and exchange symmetry links amplitudes across all labels.

Even if

$$V = \sum_{i<j}v_{ij},$$

the terms $v_{ij}$ and $v_{jk}$ generally share degree of freedom $j$. The many-body eigenproblem is therefore not a collection of isolated two-body problems.

Two-body interactions can generate higher correlations

Time evolution under a two-body Hamiltonian can generate connected three-point, four-point, and higher correlation functions. A Hamiltonian’s operator rank does not bound the correlation rank of its states.

This is one reason closure schemes matter. Mean-field and Gaussian approximations truncate or factorize a correlation hierarchy; they do not make the exact hierarchy disappear.

Momentum-Space View

For particles in a periodic volume $\Omega$, define the density mode

$$\rho_{\mathbf q} = \sum_{\mathbf k,\sigma} c_{\mathbf k+\mathbf q,\sigma}^\dagger c_{\mathbf k,\sigma}.$$

Here $\mathbf k$ and $\mathbf q$ are wave vectors, and $\rho_{\mathbf q}=\int d^d x\,e^{i\mathbf q\cdot\mathbf x}n(\mathbf x)$. For a translation-invariant pair potential with Fourier transform $\widetilde v(\mathbf q)$, the interaction can be written, in the complete periodic momentum basis, as

$$V = \frac{1}{2\Omega} \sum_{\mathbf q} \widetilde v(\mathbf q) : \rho_{\mathbf q} \rho_{-\mathbf q} :.$$

Normal ordering removes the unphysical self-pair contribution in this representation. Equivalently, in a first-quantized density identity one subtracts the $i=j$ term explicitly.

The momentum-space form makes several facts visible:

• translation invariance conserves total momentum;
• $\widetilde v(\mathbf q)$ weights wave-vector transfer $\mathbf q$ (momentum transfer $\hbar\mathbf q$);
• forward scattering, backscattering, and pairing channels can scale differently;
• long-range real-space interactions are singular or sharply structured near small momentum;
• screening and collective density response naturally appear in the same variables.

The most useful basis is therefore interaction- and observable-dependent. Real space exposes locality; momentum space exposes conservation laws, Fermi surfaces, and scattering channels.

Local Lattice Interactions

A lattice replaces continuous position by sites, orbitals, or cells. “Local” can mean onsite, finite-range, or rapidly decaying in the chosen lattice basis.

Common examples include:

$$V_{\mathrm H} = U\sum_i n_{i\uparrow}n_{i\downarrow},$$

for onsite repulsion between opposite-spin fermions;

$$V_{\mathrm B} = \frac{U}{2} \sum_i n_i(n_i-1),$$

for onsite boson repulsion;

$$V_{\mathrm{dens}} = \sum_{i<j} V_{ij}n_i n_j,$$

for intersite density interactions; and

$$V_{\mathrm{spin}} = \sum_{\langle ij\rangle} J_{ij}\, \mathbf S_i\boldsymbol\cdot\mathbf S_j,$$

for exchange-coupled spins.

These terms compete with one-body hopping, local fields, orbital splittings, and chemical potentials. In the Hubbard model, for example, the ratio of onsite interaction to kinetic bandwidth helps organize the regime, but filling, lattice geometry, temperature, and dimension remain indispensable.

Locality depends on the chosen variables

An onsite interaction in a localized orbital basis becomes a momentum-dependent four-mode scattering term after Fourier transformation. Conversely, a momentum-diagonal quadratic Hamiltonian can be long ranged in real space.

Locality is therefore a statement about both the operator and the representation. Physical locality is usually tied to spatially localized observables, but effective variables can make the same process appear nonlocal.

Constraints can be interactions in disguise

The limit $U\to\infty$ in a lattice model may forbid double occupation. Within the low-energy subspace, the interaction then appears as a kinematic constraint rather than a finite energy term. Virtual excursions outside the constrained subspace can still generate exchange and correlated hopping.

This illustrates a general principle: eliminating high-energy configurations changes both the allowed Hilbert space and the effective operators acting within it.

Short-Range and Long-Range Interactions

Interaction range is not a binary label. Distinguish at least:

• contact: idealized support only at coincident points;
• finite range: negligible beyond a microscopic distance $R$;
• short range: sufficiently rapid decay for the intended bulk and infrared arguments;
• power law: $v(r)\sim C/r^\alpha$;
• all to all: every degree of freedom couples with comparable strength;

Retardation is a separate temporal property: an effective interaction can depend on time or frequency as well as having any of these spatial ranges.

For a $d$-dimensional homogeneous system with

$$v(r) \sim \frac{C}{r^\alpha},$$

the absolute large-distance contribution, for nonzero $C$, is estimated by

$$\int_a^L dr\, r^{d-1-\alpha}.$$

Consequently:

$$\begin{array}{c|c} \alpha>d & \text{absolutely summable at large distance},\\ \alpha=d & \text{logarithmic large-distance growth},\\ \alpha<d & \text{power-law large-distance growth}. \end{array}$$

This counting test is a warning, not a complete thermodynamic theorem. Charge neutrality, screening, alternating signs, boundary shape, background terms, and conditional convergence can change the conclusion. Three-dimensional Coulomb matter is the standard reminder: the bare $1/r$ interaction is not absolutely summable, yet neutral matter can possess a stable thermodynamic limit under appropriate assumptions.

Kac normalization

For an unstructured all-to-all interaction,

$$V = J\sum_{i<j}O_iO_j,$$

there are order $N^2$ pairs. A scaling such as

$$V_{\mathrm{Kac}} = \frac{J}{N} \sum_{i<j}O_iO_j$$

keeps the operator norm at most order $N$ when the $O_i$ have a uniform bounded norm. Aligned states with a nonzero pair average attain that order, while cancellations can make other states subextensive. The rescaling does not make the interaction local or additive. Fluctuation structure, dynamics, ensemble equivalence, and finite-size scaling can still differ from short-range systems.

Range changes universality and dynamics

Long-range tails can alter:

• whether a thermodynamic limit is shape independent;
• the existence and velocity of effective light cones;
• collective-mode dispersions;
• screening behavior;
• critical exponents and upper critical dimensions;
• the validity of short-range effective field theories.

The exponent $\alpha$ alone is not enough. Sign, conservation laws, dimensionality, frustration, and the state also matter.

Microscopic, Bare, and Effective Interactions

The interaction used in a calculation is often not a fundamental two-body force.

Bare microscopic interaction

A bare interaction belongs to a specified microscopic model and regulator. Its parameters can depend on the cutoff convention. A bare contact coupling in more than one spatial dimension cannot be interpreted independently of ultraviolet regularization and matching.

Renormalized low-energy interaction

At low collision energy, a complicated short-range potential may be represented by a few scattering parameters. In a dilute three-dimensional gas, the $s$-wave scattering length $a_s$ controls the leading contact coupling. The canonical scattering definition lives in Scattering Length.

Screened interaction

Other mobile degrees of freedom can rearrange around a perturbation, converting a bare long-range interaction into a scale- and frequency-dependent screened interaction. Screening is collective: it cannot generally be assigned to one isolated pair.

Induced interaction

Eliminating phonons, photons, high-energy orbitals, or another particle species can induce interactions among the retained degrees of freedom. The result can be attractive even if a microscopic Coulomb contribution is repulsive, and it may be retarded rather than instantaneous.

Exchange as an emergent interaction

In a large-$U$ Hubbard regime, hopping through virtual doubly occupied states generates an antiferromagnetic exchange scale

$$J_{\mathrm{ex}} \sim \frac{4t^2}{U}.$$

The low-energy spin interaction is produced by statistics, hopping, and virtual high-energy configurations. Effective Hamiltonians in Many-Body Systems derives the projected model and explains its validity.

Every effective interaction should therefore carry an energy window, retained Hilbert space, and matching prescription.

Coupling Strength Must Be Dimensionless

A dimensional number is never intrinsically small. If a coupling $g$ has units, the numerical statement $g\ll1$ changes when units change.

A meaningful control parameter compares interaction and reference scales:

$$\lambda = \frac{E_{\mathrm{int}}}{E_{\mathrm{ref}}},$$

or compares lengths, densities, matrix elements, or response functions in a dimensionless combination.

There is no universal $E_{\mathrm{ref}}$. It may be a bandwidth, Fermi energy, spectral gap, level spacing, temperature, recoil energy, charging energy, or probe frequency.

System or regime	Useful control	Essential qualification
dilute stable repulsive 3D Bose condensate	$n a_s^3$, $a_s>0$	leading dilute-gas corrections scale as $\sqrt{n a_s^3}$
dilute two-component Fermi gas	$k_{\mathrm F}a_s$	resonance and effective range also matter
single-band Hubbard model	$U/W$ or $U/t$	filling, lattice, and dimension are indispensable
electron gas	$r_s$	definition depends on dimension and density convention
spin model in a field	$J/h$	sign, frustration, anisotropy, and coordination matter
Kondo impurity	$\rho(E_{\mathrm F})J_K$	logarithms grow toward low energy
long-range lattice model	interaction / local scale plus $\alpha$	normalization and thermodynamic sequence matter

The Weakly Interacting Bose Gas Preview is a clean example: weak coupling means $n a_s^3\ll1$, not merely that a symbol $g$ looks small.

Weak Coupling

A weak-coupling regime is one in which selected observables admit a controlled expansion around a specified reference problem. Schematically,

$$\mathcal O(\lambda) = \mathcal O_0 + c_1\lambda + c_2\lambda^2 + \cdots.$$

Control requires more than $|\lambda|\ll1$:

• the coefficients must not grow too rapidly at the relevant order;
• small denominators must be absent or treated by degenerate methods;
• infrared or ultraviolet logarithms must remain under control;
• the retained terms must approximate the observable over the regime used; a controlled asymptotic expansion need not converge or be analytic at the expansion point;
• system size and time must not amplify the correction beyond the nominal estimate;
• the unperturbed state must represent the correct phase and symmetry sector.

Observable dependence

The same Hamiltonian can be perturbative for one quantity and nonperturbative for another. A weak interaction may produce a small correction to a high-energy spectral moment while reorganizing the ground state at exponentially low energy.

Similarly, short-time dynamics can admit a convergent or asymptotic expansion even when late-time transport requires resummation of repeated collisions.

Degeneracy and dense spectra

The elementary mixing ratio

$$\eta_{mn} = \frac{|V_{mn}|} {|E_m^{(0)}-E_n^{(0)}|}$$

must be small for ordinary nondegenerate perturbation theory. In a many-body system, level spacings can shrink rapidly with volume. A matrix element that is microscopically small can therefore mix a dense manifold of states.

This does not make all bulk perturbation theory impossible. It means that one should expand quantities with a well-defined thermodynamic organization, use linked-cluster or Green-function methods, and resum the channels whose denominators become singular.

Large logarithms

An expression of the form

$$\lambda \ln\!\left( \frac{E_{\mathrm{UV}}}{E_{\mathrm{IR}}} \right)$$

need not be small even when $|\lambda|\ll1$. Lowering the infrared scale can invalidate every fixed-order truncation. Renormalization-group flow or channel-specific resummation may then be the controlled description.

Weak Does Not Mean Trivial

Weak attraction near a Fermi surface gives a canonical example. Repeated scattering in the Cooper channel produces a scale-dependent vertex. In a schematic convention,

$$\Gamma(E) \simeq \frac{g} {1+g\rho_{\mathrm F} \ln(E_c/E)},$$

Here $g<0$ couples time-reversed opposite-spin states, $\rho_{\mathrm F}$ is the smooth density of states per spin and volume, and $g$ has units of energy times volume. $E_c$ is the pairing-window cutoff. This schematic leading-log result assumes an unbroken pairing channel without a competing low-energy cutoff such as pair breaking. The denominator vanishes at an exponentially small scale,

$$E_* \sim E_c \exp\!\left[ -\frac{1}{|g|\rho_{\mathrm F}} \right].$$

No finite power series in $g$ produces this scale. An arbitrarily weak attraction can therefore destabilize the normal Fermi surface at sufficiently low energy under the assumptions of the pairing problem.

This is not a contradiction. The microscopic coupling is weak, while the low-energy logarithm grows until the normal-state expansion must be reorganized around a paired state.

Other weak-coupling reorganizations include:

• collective screening from repeated particle–hole processes;
• sound from repeated density interactions in a condensate;
• the Kondo scale generated by logarithmic flow;
• secular growth in nonequilibrium perturbation theory;
• bound states whose binding energy is nonanalytic in the coupling.

Intermediate coupling

When neither the interaction nor its inverse supplies a useful small parameter, exact constraints become especially valuable. A variational energy bound, a spectral sum rule, a solvable limit and agreement between independently converged numerical methods constrain different possible errors. Agreement with one of them alone does not establish accuracy for every observable.

“Strongly correlated” has no universal numerical threshold. It usually identifies a failure of a specified independent-particle, mean-field or quasiparticle description.

Strong Coupling

Strong coupling means that an interaction scale dominates a chosen competing scale. It does not mean that every quantity is uncontrolled.

Suppose

$$H = H_U+T, \qquad U\gg t,$$

where $H_U$ separates low- and high-energy local configurations by energy of order $U$, while $T$ mixes them with matrix elements of order $t$.

The low-energy theory can admit an expansion in

$$\frac{t}{U} \ll 1.$$

To second order, virtual high-energy excursions generate terms of order $t^2/U$. At half filling in the repulsive single-band Hubbard model, this yields antiferromagnetic superexchange with $J_{\mathrm{ex}}=4t^2/U$ under the standard nearest-neighbor convention.

The original model is strongly coupled relative to hopping, yet its low-energy Hamiltonian is perturbatively controlled in the inverse coupling. The correct unperturbed problem is the local interacting manifold, not the free-particle band.

Strong-coupling caveats

• A projected Hamiltonian describes only energies well below the eliminated gap.
• Degeneracy inside the low-energy manifold requires degenerate perturbation theory.
• Effective observables must be transformed along with the Hamiltonian.
• Closing the eliminated gap can invalidate the expansion. Doping instead changes the retained states: mobile holes require a model such as the $t$–$J$ model rather than a spin-only Hamiltonian, even while eliminating doublons remains controlled.
• A large bare coupling need not imply a simple low-energy state; frustration and competing exchanges may remain.

Perturbative and Nonperturbative Regimes

These terms describe analytic structure, not merely magnitude.

Perturbative contribution

A perturbative contribution is represented by a controlled series in a declared parameter, often around $\lambda=0$ or $1/\lambda=0$:

$$\mathcal O \sim \sum_{n=0}^{\infty} c_n\lambda^n.$$

The series may converge or may be asymptotic. An asymptotic series can still be highly predictive when optimally truncated and accompanied by an error estimate.

Nonperturbative contribution

A contribution such as

$$\exp\!\left(-\frac{A}{\lambda}\right)$$

has a vanishing Taylor series at $\lambda=0$ and is invisible at every finite perturbative order. Nonperturbative structure can also arise from:

• a change of phase or symmetry sector;
• tunneling between semiclassical configurations;
• topological sectors;
• bound-state formation;
• infrared flow to a new fixed point;
• constraints that reorganize the Hilbert space;
• level crossings or thermodynamic-limit nonanalyticities.

The labels are not synonyms

The following implications are false in general:

$$\begin{gathered} \text{weak coupling} \not\Rightarrow \text{all observables perturbative}, \\ \text{strong coupling} \not\Rightarrow \text{no controlled expansion}, \\ \text{nonperturbative} \not\Rightarrow \text{numerically large}. \end{gathered}$$

An exponentially small gap is nonperturbative but small. A leading strong-coupling exchange can be perturbative in $t/U$ and physically decisive.

Crossovers, Instabilities, and Phases

A coupling ratio can mark several different phenomena.

Crossover

A crossover is a smooth change in the most useful description. No thermodynamic-limit nonanalyticity is required. A gas can cross from classical to degenerate statistics, or a quasiparticle can evolve from long lived to overdamped.

Instability of a reference state

A susceptibility divergence or negative mode can show that the chosen reference state is unstable. This indicates the need for a reorganized state or effective theory, but it does not by itself determine the final phase.

Phase transition

A sharp phase transition is a bulk statement involving nonanalytic thermodynamic-limit behavior, a change of symmetry realization, topological structure, or another phase diagnostic. Quantum Phase Transitions explains the zero-temperature definition and finite-size limitations.

One should not infer a phase transition merely because a perturbative approximation fails. The method may fail before, at, or after the physical transition, and it can fail even in a single smooth phase.

Competing Preferred Bases

Many-body difficulty often comes from noncommuting terms that prefer incompatible states.

For a lattice Hamiltonian

$$H = H_{\mathrm{kin}}+H_{\mathrm{int}},$$

the kinetic term may be diagonal in momentum occupation, while the interaction is simplest in localized occupation. If

$$[H_{\mathrm{kin}},H_{\mathrm{int}}] \ne 0,$$

there is no complete common eigenbasis for finite-dimensional Hermitian terms. They can nevertheless share an individual eigenvector or even a ground state; noncommutation alone does not prove competing minima.

The transverse-field Ising model presents the same structure in spin language: exchange favors $z$-ordered configurations, while the transverse field favors $x$ polarization. The competition, not merely the size of either matrix, produces quantum fluctuations and possible criticality.

This observation guides approximations. Start from the term that defines a controlled manifold, but test whether the neglected term mixes nearby states strongly.

Correlations as the Signature of Interaction

For observables $A$ and $B$, define the connected correlation

$$C_{AB} = \langle AB\rangle - \langle A\rangle \langle B\rangle.$$

Interactions often generate nonzero connected correlations, but several qualifications matter:

• exchange statistics produces correlations even in noninteracting gases;
• constraints can correlate occupations without an explicit finite interaction term;
• a product mean-field state sets selected connected correlations to zero by construction;
• a vanishing two-point connected correlator does not imply absence of higher correlations;
• correlation is not automatically entanglement.

Correlation Functions Overview explains the different correlators, their orderings and Fourier conventions, and their relation to measured response.

Coupling derivatives as diagnostics

For

$$H(g) = H_0+gV,$$

the Hellmann–Feynman relation for a nondegenerate normalized eigenstate gives

$$\frac{dE_n}{dg} = \langle n(g)|V|n(g)\rangle.$$

At finite temperature, with

$$F(g) = -\frac{1}{\beta} \ln\operatorname{Tr} e^{-\beta H(g)},$$

one similarly has

$$\frac{\partial F}{\partial g} = \langle V\rangle_{\beta,g}.$$

These identities connect energy or free-energy changes directly to interaction observables. They are useful checks for analytic approximations, finite-difference calculations, and numerical simulations.

Mixing is measured relative to a gap

An exactly solvable two-state block makes the meaning of “small” concrete. Let

$$H=\begin{pmatrix}0&v\\v^*&\Delta\end{pmatrix}, \qquad \Delta>0,\qquad \eta=\frac{|v|}{\Delta}.$$

The lower eigenstate has probability

$$p=\frac12\left(1-\frac{1}{\sqrt{1+4\eta^2}}\right)$$

in the second, initially higher-energy basis state. For $\eta\ll1$, $p=\eta^2+O(\eta^4)$. When the gap becomes comparable to the matrix element, the second state cannot be treated as a small correction. In a many-body calculation, this motivates retaining a nearly degenerate subspace and diagonalizing within it.

The relevant comparison is $|v|/\Delta$. The dashed quadratic estimate is useful near the origin; extrapolating it to order-one mixing fails even though the exact two-state problem remains simple.

Structure-Matched Starting Methods

Structure of the problem	Natural starting route	Main validation question
weak coupling, no singular channels	many-body perturbation theory	are corrections and residuals small?
weak coupling with large logarithms	resummation or renormalization group	which channel flows first?
condensate with small depletion	Gross–Pitaevskii plus Bogoliubov fluctuations	is depletion parametrically small?
product-like order or large coordination	self-consistent mean field	are connected fluctuations controlled?
fermionic Slater-determinant regime	Hartree–Fock	are correlation energies and instabilities small?
dominant local interaction with a gap	projection and strong-coupling expansion	is the eliminated sector well separated?
controlled large parameter	large-N and saddle-point methods or semiclassics	are finite-parameter corrections bounded?
one-dimensional low-entanglement ground state	matrix-product-state methods	are bond dimension and finite size converged?
nonnegative statistical weights	quantum Monte Carlo	are finite-size and continuation errors controlled?
small symmetry-resolved Hilbert space	exact diagonalization or Krylov methods	do observables scale consistently with size?
no evident small parameter	cross-method benchmarking	which independent constraints agree?

The method should be selected after naming the observable. Ground-state energy, order parameter, high-frequency moment, low-frequency conductivity, long-time relaxation, and entanglement growth can demand different approximations for the same Hamiltonian.

Exercises

Power-law summability

For $v(r)\sim C/r^\alpha$ in $d$ dimensions, evaluate the large-$L$ behavior of

$$I(L) = \int_a^L dr\, r^{d-1-\alpha}.$$

Solution

For $\alpha\ne d$,

$$I(L) = \frac{ L^{d-\alpha} - a^{d-\alpha} }{d-\alpha}.$$

Thus $I(L)$ converges as $L\to\infty$ when $\alpha>d$ and grows as $L^{d-\alpha}$ when $\alpha<d$. At $\alpha=d$,

$$I(L) = \ln\!\left(\frac{L}{a}\right).$$

This establishes the absolute-summability test. Neutrality, oscillating signs, screening, and boundary prescriptions can still modify the thermodynamic conclusion.

A weak-coupling nonperturbative scale

Let

$$E_*(\lambda) = E_c e^{-1/\lambda}, \qquad \lambda>0.$$

Show that every derivative of $E_*$ at $\lambda=0^+$ vanishes, even though $E_*>0$ for every $\lambda>0$.

Solution

Each derivative has the form

$$\frac{d^n E_*}{d\lambda^n} = E_c e^{-1/\lambda} P_n\!\left(\frac{1}{\lambda}\right),$$

where $P_n$ is a polynomial. The exponential decays faster than any inverse power grows, so

$$\lim_{\lambda\to0^+} \frac{d^n E_*}{d\lambda^n} = 0$$

for every finite $n$. The Taylor series about zero is therefore identically zero and cannot reproduce $E_*$ at positive coupling. The scale is nonperturbative in $\lambda$ even though it is exponentially small.

Failure of a naive small-coupling test

Two unperturbed states have spacing $\Delta$ and are coupled by a matrix element $v$. The Hamiltonian in that subspace is

$$H = \begin{pmatrix} 0 & v\\ v^* & \Delta \end{pmatrix}.$$

Find the dimensionless mixing parameter and state when nondegenerate perturbation theory fails.

Solution

The eigenvalues are

$$E_\pm = \frac{\Delta}{2} \pm \sqrt{ \frac{\Delta^2}{4} + |v|^2 }.$$

The relevant ratio is

$$\eta = \frac{|v|}{|\Delta|}.$$

For $\eta\ll1$, ordinary nondegenerate corrections are controlled. When $\eta$ is order one or larger, the states mix strongly and the two-dimensional block should be diagonalized exactly. A small dimensional value of $v$ is irrelevant if the spacing is even smaller.

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